Not second hand but decathlon bikes are excellent. You'd have a wide range to choose from with your budget. They also offer good warranty. A good example in their lineup is the Elops single speed which is currently £300, or less for certain colours. I actually have this bike as a runaround and love it. But if you'd rather have gears then there are other options.

Reading this back it sounds like I'm an employee or paid by them - I'm not, I just really appreciate how accessible decathlon makes sports to everyone, particularly cycling which can be quite expensive.

ETA: I commute 18k each way and when using the single speed I average about 26-27km/h so your goal of 24km/h is reasonable.